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DragosP
Pe: 2018-12-24T06:14:29+02:00 In: In:

Limita din integrala UTCN

\lim_{n\to inf}\int_{1/n}^{n}\frac{arctg(x^2)}{1+x^2}
Raspunsul ar trebui sa dea \frac{\Pi ^{2}}{8}.
Nu am deloc inspiratie cum sa procedez la acest exercitiu.

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3 raspunsuri

  1. Felixx
    veteran (III)

    Faci substitutia de variabila x=\frac{1}{t},apoi folosesti relatia cunoscuta:
    arctgx+arctg\frac{1}{x}=\frac{\pi }{2}
    Apoi „rupi”integrala in doua.A doua iti va da -I_{n}, iar la prima tii cont de relatia:
    arctga-arctgb=\frac{a-b}{1+ab}
    Eu am obtinut :
    I_{n}=\frac{\pi }{4}arctg\frac{n^{2}-1}{2n}

  2. DragosP

    Multumesc enorm!

  4. Felixx
    veteran (III)

    Felixx post_id=112586 time=1545648679 user_id=21270 wrote:
    Faci substitutia de variabila x=\frac{1}{t},apoi folosesti relatia cunoscuta:
    arctgx+arctg\frac{1}{x}=\frac{\pi }{2}
    Apoi „rupi”integrala in doua.A doua iti va da -I_{n}, iar la prima tii cont de relatia:
    arctga-arctgb=\frac{a-b}{1+ab}
    Eu am obtinut :
    I_{n}=\frac{\pi }{4}arctg\frac{n^{2}-1}{2n}


    OBS.
    Scuze ,o mica corectie:
    arctga-arctgb=arctg\frac{a-b}{1+ab}

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